The wavefunctions, probability densities and shape of

The wavefunctions, probability densities and shape of 2s atomic orbitals... 

The wavefunction of an electron associated with an atomic nucleus. The orbital is typically depicted as a three-dimensional electron density cloud. If an electron s azimuthal quantum number (/) is zero, then the atomic orbital is called an s orbital and the electron density graph is spherically symmetric. If I is one, there are three spatially distinct orbitals, all referred to as p orbitals, having a dumb-bell shape with a node in the center where the probability of finding the electron is extremely small. (Note For relativistic considerations, the probability of an electron residing at the node cannot be zero.) Electrons having a quantum number I equal to two are associated with d orbitals. 

Our individual one-electron HF or KS wavefunctions represent the individual molecular orbitals, and the square of the wavefunction gives us the probability distribution of each electron within the molecule. We do not know the form of the real multi-electron wavefunction a priori, nor the individual one-electron HF or KS functions, but we can use the mathematical principle that any unknown function can be modeled by a linear combination of known functions. A natural choice for chemists would be to use a set of functions that are similar in shape to individual atomic orbitals. To do this, we need to consider atomic radial distribution functions, such as the ones shown in Figure 3.2 for hydrogen. These are plots of how the electron density varies at any given distance away from the nucleus. 

Representation of several of the squared spherical harmonic functions with m=0. The shapes depict the surface of a contour drawn for some fixed probability density ampHtude, and the alternation in shading designates regions of different phases (different signs) of the wavefunction. 

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